Game Development Reference
In-Depth Information
is that it boiled down decision making into small enough parts that it is easy to
discern the concepts in play. Examples in game theory are often characterized by
simple rules and equally transparent decision possibilities. (The actual proofs of
von Neumann and Morgenstern's assertions of game theory are not even remotely
as compact—to the point that
Theory of Games and Economic Behavior
is largely
unreadable.) John von Neumann starts small—with the concept of the two-player,
zero-sum, perfect knowledge game and works up from there.
Through this reverence to Occam's razor, we are given the smallest possible
building blocks from which we can learn and with which we can build later. Using
these examples, we can establish a foundation of understand about what goes into
decisions, not only from a logical standpoint, but from a mathematical one as well.
It is this application of mathematics to decision theory that allows us to model the
subtlety to which I referred in Chapter 1.
M
ATCHING
P
ENNIES
In Chapter 4 of
Theory of Games and Economic Behavior,
von Neumann and
Morgenstern begin with a section entitled “Some Elementary Games,� which has
a subsection, “The Simplest Games.� They weren't kidding. The first one they
introduce is one not so obscurely called Matching Pennies. It is similar to Rock-
Paper-Scissors but with two choices instead of three. (I told you they weren't
kidding!) It is a zero-sum game in that one player's loss is another's gain.
The game involves two players and two pennies. (I grudgingly use the term
game
despite the lack of interesting choices… but that's just my two cents.) Each player
hides his penny and turns it to either heads or tails. They then simultaneously show
each other their pennies, and the enormously complex ordeal of scoring the round
ensues. As you can see in Figure 5.1, if the pennies match—either both heads or
both tails—then player A is the winner. Conversely, if the coins do not match, then
player B is the winner. (I'm not sure how it is determined who will be player A, but
I would suggest that they could flip a coin.)
The lesson to be learned here is strictly one of terminology. Matching Pennies
is a game in which there is no “pure strategy.� That is, there is no “best response.�
A
pure strategy
is a complete definition of what a player
should
do at any given
time. That is,
if
you play this way, you will win (or at least will maximize your
chances of doing so). For example, as we discussed before, Tic-Tac-Toe has a pure
strategy. If you want to win, you have to make certain moves. If you make those
moves, you maximize your chances—which, in the case of Tic-Tac-Toe, generally
leads to a draw. At worst, it doesn't lead to a loss. At best, if your opponent makes
a mistake (by not playing the pure strategy), you will win.
